The matter is to model the physical behavior of the loudspeaker to simulate the operation thereof when the audio signal is applied thereto after amplification, so that various corrective processing operations can be performed upstream on this audio signal in order to optimize the quality of the final acoustical reproduction rendered to the listener.
In particular, it is current to reinforce the low frequencies to compensate for the fact that the loudspeakers dedicated to this register, or woofers, which are generally installed in open (vent system) or closed baffles, are always more or less limited in the rendering of the deepest frequencies, the low limit (referred to as the baffle cut-off frequency) depending on the size of the loudspeaker, the volume of the baffle and the type of mounting used.
However, if the level of the electrical signal is increased in the low frequencies by a suitable, analog or digital, filtering, the excursion of the loudspeaker diaphragm, i.e. the amplitude of its displacement with respect to its equilibrium position, becomes rapidly too high, with a risk of damaging the loudspeaker, and, at the very least, the introduction, for excessive excursion values, of distortions, clippings and saturations that rapidly deteriorate the rendering quality of the audio signal.
Knowing the overall response of the loudspeaker allows anticipating this risk, to limit if need be the level of the signal to be reproduced in order to avoid excessive excursions or nonlinearities that generate distortions. Another type of conceivable processing consists in applying to the audio signal a specific filtering for compensating for the nonlinearities introduced by the loudspeaker, so as to reduce the audio distortions and to provide a better listening quality.
The matter is then, independently of any limitation of the maximal excursion, to make the loudspeaker diaphragm displacement the more linear possible, in particular for the deepest frequencies, by compensating for the physical limitations of the loudspeaker response in this register, in the vicinity and below the acoustical cut-off frequency of the loudspeaker/baffle unit.
Knowing the parameters that model the overall response of the loudspeaker is essential to perform such processing operations.
These parameters are conventionally those referred to as “Thiele and Small” (T/S), which describe a modeling of an electrodynamic loudspeaker taking into account the various electrical, mechanical and acoustical phenomena involved in the reproduction of the signal, as well as the electromechanical and mechanical-acoustical conversions. The loudspeaker response, in particular for the low frequencies, may then be described by a set of parameters, uniformly referenced by the loudspeaker manufacturers.
These T/S parameters are however not constant in time, nor linear.                firstly, they are liable to drift over time, as a function for example of the loudspeaker ageing, the heating during use, etc.;        secondly, if it is desired to have a precise and realistic modeling of the loudspeaker behavior, it must be taken into account that some of these parameters are not linear, i.e. their values are not fixed but varies constantly as a function of the instantaneous excursion, i.e. the position at a given instant of the loudspeaker moving coil and diaphragm with respect to the central equilibrium position. It is in particular the case for the electrical inductance, the total mechanical stiffness of the system (the stiffness of the diaphragm increasing as the latter goes away from its equilibrium position) and the diaphragm driving “force factor” (linked to the magnetic field of the coil gap, it decreases as the coil goes away from the equilibrium position).        
The EP 1 799 013 A1 describes a technique for predicting the behavior of a loudspeaker, based on the T/S parameters, so as to compensate for the nonlinearities of the loudspeaker and to reduce the audio distortions introduced in the acoustic signal rendered to the user.
The T/S parameters are however considered therein as invariants, which are known a priori, so that the response modeling is fixed and cannot take into account the slow evolutions of the parameters, dues for example to their drift over time on account of the ageing of the components.
The US 2003/0142832 A1 describes a technique of adaptive estimation of the parameters of a loudspeaker, including nonlinear parameters, based on the measurement of the current through this loudspeaker, with implementation of a gradient descent algorithm. This method requires a previous determination of the parameters during a static calibration phase: during this calibration, the T/S parameters are calculated for various position values of the diaphragm (offset with respect to the equilibrium position), with measurement of the impedance. Thereafter, a measurement of the current is compared to an estimation of this same current (squared and filtered by a low-pass filter) to calculate the derivative of the error with respect to each parameter. The technique also implements a gradient descent algorithm, of the Least Means Square (LSM) type.
This method however suffers from the drawback that it requires a previous calibration phase with impedance measurements and application of a predetermined signal, which excludes a re-estimation of the subsequent parameters, anyway by a general public user. On the other hand, the simple algorithms of the gradient-descent LMS type do not take into account the measurement noises, which are inevitable, so that the estimator is rather little efficient in real cases of use.
The US 2008/0189087 A1 describes another technique of estimation of the parameters of a loudspeaker, also of the gradient descent LMS type. More particularly, the method processes separately the estimation of the linear part and that of the nonlinear part. For that purpose, the error signal used by the LMS algorithm (difference between the measured signal and the predicted signal) is processed so as to decorrelate the linear part from the nonlinear part. This document also proposes to implement the estimator by applying at the input a particular audio signal, modified by a comb filter that selectively eliminates certain chosen frequencies.
This technique has the same drawbacks as the previous one, in particular the necessity of a calibration based on a modified input signal liable to impair the comfort of listening of the user, which does not allow performing the estimation during music listening, in a transparent manner for the user. Still another method is described in the university paper of Marcus Arvidsson and Daniel Karlsson, Attenuation of Harmonic Distorsion in Loudspeakers Using Non-Linear Control, Department of Electrical Engineering, Linköopings Universitet (SE), dated 18 Jun. 2012, XP055053802. This method is based on an observation vector that comprises only measurements of electrical parameters (voltage and current), which are applied to an extended Kalman predictive filter estimator. This estimator performs the prediction of a state vector whose components comprise the value of the excursion and the value of the current in the loudspeaker. But this method does not allow estimating on-the-fly both the linear and nonlinear parameters of the loudspeaker response to thereafter apply a suitable corrective audio processing.